2.15.11 Degasperis Procesi \(u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx}\)

problem number 120

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Degasperis Procesi. Solve for \(u(x,t)\)

\[ u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx} \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] - D[D[u[x, t], {x, 2}], t] + 4*u[x, t]*D[u[x, t], x] == 3*D[u[x, t], x]*D[u[x, t], {x, 2}] + u[x, t]*D[u[x, t], {x, 3}]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(u(x,t),t)-diff(u(x,t),x,x,t)+4*u(x,t)*diff(u(x,t),x)=3*diff(u(x,t),x)*diff(u(x,t),x$2)+u(x,t)*diff(u(x,t),x$3); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
 

\[u \left (x , t\right )=\frac {f_{3} \left (x \right )}{-t \textit {\_c}_{2}+c_{2}}\boldsymbol {\operatorname {where}}\left [\left \{\left \{f_{3} \left (x \right )=\textit {\_a} \boldsymbol {\operatorname {where}}\left [\left \{\frac {\left (\frac {d^{2}}{d \textit {\_a}^{2}}\textit {\_}b\left (\textit {\_a} \right )\right ) \textit {\_}b\left (\textit {\_a} \right )^{2} \textit {\_a} +\left (\frac {d}{d \textit {\_a}}\textit {\_}b\left (\textit {\_a} \right )\right )^{2} \textit {\_}b\left (\textit {\_a} \right ) \textit {\_a} +\textit {\_}b\left (\textit {\_a} \right ) \left (\frac {d}{d \textit {\_a}}\textit {\_}b\left (\textit {\_a} \right )\right ) \left (\textit {\_c}_{2}+3 \textit {\_}b\left (\textit {\_a} \right )\right )-\textit {\_a} \left (\textit {\_c}_{2}+4 \textit {\_}b\left (\textit {\_a} \right )\right )}{\textit {\_a}}=0\right \}, \left \{\textit {\_a} =f_{3} \left (x \right ), \textit {\_}b\left (\textit {\_a} \right )=\frac {d}{d x}f_{3} \left (x \right )\right \}, \left \{x =\int \frac {1}{\textit {\_}b\left (\textit {\_a} \right )}d \textit {\_a} +c_{1} , f_{3} \left (x \right )=\textit {\_a} \right \}\right ]\right \}\right \}\right ]\]

But still has unresolved ODE’s in solution

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